While not obvious from its initial motivation in linear algebra, there are many context where iterated traces can be defined. In this paper we prove a very general theorem about 2-categorical traces. We show that Lefschetz-type theorems the literature consequences of result and new perspective provide allows for immediate spectral generalizations.

We construct an $(\infty,2)$-version of the (lax) Gray tensor product. On 1-categorical level, this is a binary (or more generally $n$-ary) functor on category $\Theta_2$-sets, and it shown to be left Quillen with respect Ara's model structure. Moreover we prove that product forms part "homotopical" (biclosed) monoidal structure, or precisely normal lax structure associative up homotopy.